Dynamic Behavior Of Two Types Of Reaction-diffusion Model

The development of natural science relies largely on the progress in physics,chem-istry,life sciences and so on.The mathematization of these concrete subjects is important to make further research.Many mathematical models can be naturalized as the reaction-diffusion models.In recent decades,the research of reaction-diffusion models have made great progress.With the deepening of the research,the reaction-diffusion models are widely used to explore the great mass of dynamic systems with diffusion.By means of the nonlinear analysis and the theories of nonlinear partial differential equations,this thesis will study the dynamic behavior of two types of reaction-diffusion model.The main research contents include a priori estimates,non-existence,existence(bifurcation structure),uniqueness,stability and asymptotic behavior of positive equilibrium solution-s.The methods used here involve the maximum principle,energy method,implicit func-tion theorem,bifurcation theory,topological degree theory,stability theory,regularization theory,perturbation theory and Lyapunov-Schmidt reduction.This thesis includes the fol-lowing three aspects:Chapter 1 presents firstly the research background and present situation of the reaction-diffusion model with the Degn-Harrison reaction scheme,and the Leslie predator-prey model with cross-diffusion and protection zone.Next,the main works of this thesis are introduced.In Chapter 2,a reaction-diffusion model with Degn-Harrison reaction scheme is con-sidered.Some fundamental analytic properties of nonconstant positive solutions are first investigated.Next,the stability of constant steady-state solution to both ODE and PDE models is studied.Our result also indicates that if either the size of the reactor or the ef-fective diffusion rate is large enough,then the system does not admit nonconstant positive solutions.Finally,we establish the global structure of steady-state bifurcations from sim-ple eigenvalues by bifurcation theory and the local structure of the steady-state bifurca-tions from double eigenvalues by the Lyapunov-Schmidt reduction and implicit function theorem.In Chapter 3,we study the change of behavior of positive solutions in a Leslie predator-prey model when a simple protection zone and cross-diffusion for the prey are introduced.We first analyze the effects of cross-diffusion and protection zone on the bi-furcation continuum of positive solutions.Moreover,the asymptotic behaviors of positive solutions as certain parameters are large or small are discussed.Finally,for small birth rates of two species and large cross-diffusion for the prey,the detailed structure and sta-bility of positive solutions are established.Our results indicate that the environmental heterogeneity,together with large cross-diffusion,can produce much more complicated s-tationary patterns,moreover,these results are quite different from those of Lotka-Volterra models.